Neutrino Masses and Dark Matter Stability in 3HDMs with Minimal Non-Abelian Discrete Symmetries

Abstract: We present minimal three-Higgs-doublet models (3HDMs) based on global non-Abelian discrete symmetries that simultaneously explain neutrino masses and dark matter stability. A residual $Z_2$ parity from the spontaneous breaking of the new symmetry stabilizes the dark matter candidate, which runs in the loop generating neutrino masses at one loop alongside a tree-level type-I seesaw contribution. We identify $S_3$ and $D_4$ as the smallest non-Abelian groups realizing this framework and determine the minimal models in both cases that are consistent with current neutrino oscillation data. The resulting models conserve CP in both the Yukawa and scalar sectors.

• The paper introduces minimal three-Higgs-doublet models using S3 and D4 symmetries to generate neutrino masses via type-I seesaw and one-loop radiative corrections while ensuring dark matter stability through an emergent Z2 symmetry.
• The S3 variant yields a rank-2 neutrino mass matrix with one massless eigenstate and effective Majorana mass in the 1.45–3.70 meV range, whereas the D4 model produces fully massive neutrinos with a broader parameter space.
• Extensive parameter scans confirm compatibility with oscillation data, collider limits, and relic density constraints, making these models promising for future tests in lepton flavor experiments and dark matter searches.

Neutrino Masses and Dark Matter Stability in 3HDMs with Minimal Non-Abelian Discrete Symmetries

Motivation and Theoretical Framework

The work systematically builds minimal three-Higgs-doublet models (3HDMs) that address two central problems in fundamental physics: the mechanism of neutrino mass generation and the stability of dark matter (DM). The framework imposes global non-Abelian discrete symmetries—specifically, $S_3$ and $D_4$, the smallest such groups with two-dimensional irreducible representations—as organizing principles. The approach embodies the Discrete Dark Matter (DDM) mechanism: the stabilizing $Z_2$ symmetry of DM is not put in by hand but emerges as a remnant of flavor symmetry breaking. This connection enables the simultaneous resolution of neutrino mass flavor structure and DM stability in an economical way.

In these models, the tree-level atmospheric neutrino mass scale arises via a type-I seesaw with right-handed (RH) neutrinos, while the solar scale is generated radiatively at one-loop, mediated by particles in the "dark" sector that are stabilized by the residual $Z_2$ symmetry. The scalar sector consists of three $SU(2)_L$ doublets: a Standard Model–like singlet $H_1$ and a $G_D$ doublet $\Phi = (H_2, \eta)^T$, with one vevless component ($\eta$) serving as the DM candidate. The minimal fermion content includes a pair of RH neutrinos $N_D = (N_1, N_2)^T$ in the doublet representation and—crucially for full phenomenological viability—an additional singlet $D_4$0.

Mechanism of Neutrino Mass Generation

The neutrino sector is constructed such that $D_4$1 produces a tree-level, rank-1 mass matrix (type-I seesaw), accounting for the atmospheric mass scale. The second scale is generated radiatively at one-loop via interactions involving $D_4$2 and the inert scalar $D_4$3, which is stabilized by the residual $D_4$4. This mechanism is depicted in

Figure 1: Neutrino mass contributions at tree and loop level as in~[Rojas:2018wym, Aranda:2018lif].

The combined contributions reproduce two non-zero, hierarchical mass splittings consistent with data. However, achieving alignment with all lepton mixing parameters and mass splittings necessitates $D_4$5, a RH neutrino singlet under $D_4$6.

Model Structures: $D_4$7 and $D_4$8 Realizations

$D_4$9-Based 3HDMs

The $Z_2$0 scenario assigns all leptons as singlets and $Z_2$1 as a doublet. The resulting neutrino mass matrix is generically rank-1 from the doublet sector and is lifted to rank-2 by the singlet $Z_2$2. This structure strictly enforces one massless neutrino, $Z_2$3, and two massive states, directly linking the model's output to the measured oscillation mass squared differences, with no freedom for a nontrivial lightest neutrino mass. The charged lepton sector features sufficient non-diagonality to account for lepton mixing. Numerical scans identify parameter sets fully compatible with current global fit data, with mixing angles and splittings within $Z_2$4 of best-fit values.

A critical phenomenological prediction is the suppression of neutrinoless double beta decay, with the effective Majorana mass confined to $Z_2$5–$Z_2$6, well beneath the reach of current experiments but possibly accessible to future endeavors.

$Z_2$7-Based 3HDMs

In $Z_2$8-based models, the extended singlet content allows the construction of fully rank-3 neutrino mass matrices. Depending on the assignments, multiple Yukawa textures emerge, three of which yield viable three-flavor mixing. Unlike the $Z_2$9 case, the $Z_2$0 mass matrix does not enforce $Z_2$1; all three neutrinos acquire mass, and the effective Majorana mass for $Z_2$2 decay is generically higher, in the $Z_2$3–$Z_2$4 range for $Z_2$5. The parameter space accommodates all five oscillation observables within $Z_2$6 of the global fit.

The charged scalar, CP-even, and CP-odd scalar mass eigenstates all satisfy direct collider bounds (masses $Z_2$7 400 GeV), and the viable dark matter candidate—$Z_2$8—can realize the observed cosmological relic density in standard thermal scenarios for $Z_2$9.

CP Conservation and Theoretical Constraints

A notable feature of both model variants is exact CP conservation in the Yukawa and scalar sectors, simplifying the viable parameter space. All relevant quartic couplings are chosen real, and theoretical constraints (perturbativity, vacuum stability via copositivity, boundedness from below) are enforced throughout. The mixing of the SM-like scalar is constrained to match LHC measurements, and all points satisfy strong collider limits, including those on new scalars and heavy fermions.

A random scan over the relevant parameter space—Yukawas, heavy fermion masses ($SU(2)_L$0 GeV), and scalar quartics—combined with the outlined theoretical and collider constraints, yields points with exact agreement to oscillation data. The inert-doublet relic density analysis points to two allowed DM mass windows, with model benchmarks residing in the upper window and additional annihilation channels potentially further reducing the predicted relic abundance.

Implications and Prospects

The paper demonstrates that the DDM mechanism can be realized with a flavor symmetry group smaller than $SU(2)_L$1, reducing the Higgs sector from four doublets (in minimal $SU(2)_L$2 constructions) to three, and without requiring explicit CP violation in the scalar sector. This economy in scalar content, flavor group size, and parameter space makes these 3HDMs an appealing paradigm for constructing models linking neutrino masses to DM stability via flavor symmetry breaking.

From a phenomenological perspective, the models predict suppressed rates of $SU(2)_L$3 at or below the $SU(2)_L$4 scale, a nearly massless lightest neutrino in the $SU(2)_L$5 scenario, and viable WIMP DM candidates consistent with inert doublet phenomenology. The tight structure correlates the neutrino spectrum and mixing angles with the presence and mass of new electroweak-scale scalars and fermions, opening a route for experimental testing at future lepton flavor experiments, dark matter searches, and next-generation neutrinoless double beta decay experiments.

Conclusion

Minimal 3HDMs based on the non-Abelian discrete groups $SU(2)_L$6 and $SU(2)_L$7 provide a tightly predictive and phenomenologically viable framework wherein dark matter stability and the structure of neutrino masses are consequences of the same fundamental symmetry principle. The models enforce strong numerical agreement with current lepton flavor and DM constraints, achieve exact CP conservation, and provide direct phenomenological targets for future intensity and energy frontier probes. The reduction to three Higgs doublets relative to previous constructions is significant for both theoretical economy and collider phenomenology. Further studies, particularly those exploring thermal relic abundance and direct or indirect detection channels for the DM candidate, as well as possible leptogenesis implications, are motivated by the structures uncovered here.